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7.2
Areas in the Plane
Gateway Arch, St. Louis, Missouri
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly,
2003
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1
2
y
x
= 
2
y
x
= 
How can we find the area
between these two curves?
We could split the area into
several sections, use
subtraction and figure it out,
but there is an easier way.
→
2
1
2
y
x
= 
2
y
x
= 
Consider a very thin vertical
strip.
The length of the strip is:
1
2
y
y

or
( 29
( 29
2
2
x
x

 
Since the width of the strip is
a very small change in
x
, we
could call it
dx
.
→
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1
2
y
x
= 
2
y
x
= 
1
y
2
y
1
2
y
y

dx
Since the strip is a long thin
rectangle, the area of the strip is:
( 29
2
length
width
2
x
x dx
⋅
=

+
If we add all the strips, we get:
2
2
1
2
x
x dx


+
∫
→
2
1
2
y
x
= 
2
y
x
= 
2
2
1
2
x
x dx


+
∫
2
3
2
1
1
1
2
3
2
x
x
x


+
8
1
1
4
2
2
3
3
2

+
  +
+
8
1
1
6
2
3
3
2

+ 

36 16 12
2 3
6

+
 
27
6
=
9
2
=
→
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( 29 ( 29
1
2
Area
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This note was uploaded on 02/12/2011 for the course MAC 2233 taught by Professor Smith during the Spring '08 term at University of Florida.
 Spring '08
 Smith

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